What can we say about the convergent subsequences of $\sin(n^2)$ whose existence is guaranteed by the Bolzano-Weierstrass theorem?
Can we, as a corollary, claim that for all $\epsilon \gt 0$ there exists $m,n \in \mathrm{N}$ such that $|\sin(n)-\sin(m)|\lt \epsilon$ ?
For any $r\in [-1,1]$ for any $\epsilon > 0$ there are infinitely many integers $n$ such that $|\sin n^2 - r| < \epsilon.$
The answer to the second question is "yes" if you can prove that your subsequence is Cauchy, which it is because it is convergent. The proof of this uses the triangle inequality which is somewhat simple so I'll let you tackle that :)
Based on the periodicity of the sine function, perhaps there might be quite a few limit points in the range $[-1,1]$ of the sequence $(\sin(n^2))_{n = 1}^{\infty}$.
